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Stability of Constrained Capillary Surfaces FL47CH22-Steen ARI 1 December 2014 20:31 Stability of Constrained Capillary Surfaces J.B. Bostwick1 and P.H. Steen2 1Department of Engineering Science and Applied Mathematics, Northwestern University, Evanston, Illinois 60208 2School of Chemical and Biomolecular Engineering and Center for Applied Mathematics, Cornell University, Ithaca, New York 14853; email: [email protected] Annu. Rev. Fluid Mech. 2015. 47:539–68 Keywords First published online as a Review in Advance on surface tension, wetting, drops, bridges, rivulets, contact line, common line September 29, 2014 Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org The Annual Review of Fluid Mechanics is online at Abstract fluid.annualreviews.org A capillary surface is an interface between two fluids whose shape is deter- Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. This article’s doi: mined primarily by surface tension. Sessile drops, liquid bridges, rivulets, and 10.1146/annurev-fluid-010814-013626 liquid drops on fibers are all examples of capillary shapes influenced by con- Copyright c 2015 by Annual Reviews. tact with a solid. Capillary shapes can reconfigure spontaneously or exhibit All rights reserved natural oscillations, reflecting static or dynamic instabilities, respectively. Both instabilities are related, and a review of static stability precedes the dynamic case. The focus of the dynamic case here is the hydrodynamic sta- bility of capillary surfaces subject to constraints of (a) volume conservation, (b) contact-line boundary conditions, and (c) the geometry of the supporting surface. 539 FL47CH22-Steen ARI 1 December 2014 20:31 1. OVERVIEW Capillary surfaces are at once a modern and classical topic. Shape reconfiguration is central to Capillary surface: a emerging applications that exploit the physics of liquids confined by surface tension on the micro- mathematical surface and nanoscale, whereas the study of shape stability reaches back to the 1800s (see the sidebar between two fluids endowed with surface Historical Perspective). We review recent progress in the dynamic stability analysis of surfaces, tension focusing on the influence of constraint by solid supports. Volume (pressure) An early stability prediction is due to Plateau (1863), who reported the instability to volume dis- disturbance: turbances of a capillary cylinder for lengths longer than its circumference, the well-known Plateau disturbance that limit. Some years later, Rayleigh (1879) calculated the growth rate of instability as it depends on preserves the volume the disturbance wave number to estimate the final droplet size for a liquid jet disintegrating into (pressure) of the base droplets. The Plateau limit is recovered from Rayleigh’s solution of the hydrodynamic problem by state putting the growth rate to zero [the Plateau limit is sometimes referred to as the Plateau-Rayleigh Plateau-Rayleigh (PR) limit]. This illustrates how static stability can be recovered from a dynamic analysis and (PR) limit: the length, equal to its provides the prototypical relationship between the stabilities, a theme of this review. base-state Surface tension largely determines the surface shape on scales smaller than the capillary length 1/2 circumference, beyond lc ≡ (σ/ρg) ,whereρg is the buoyant force per volume according to the gravity level g and the which a cylindrical density difference between the fluids ρ. For fluids separated by a thin film (e.g., soap films) or a capillary surface is fluid against an immiscible fluid (e.g., Plateau tanks), the density difference ρ can be small, yielding unstable to volume disturbances tens-of-centimeter capillary lengths. In space applications, the capillary length for a liquid against gas can be 100 cm, whereas it is usually a few millimeters on Earth. Plateau tank: a chamber that enables Michael (1981) gave a then-timely snapshot of the field in this journal, with an emphasis large lc on Earth by on static stability and methods. Interest in shape stability, with its enhanced role in low-gravity immersing a capillary fluid mechanics, grew as space programs grew. Myshkis et al. (1987) provided a comprehensive surface in an monograph with this motivation. Also informed by capillarity in weightlessness is Langbein (2002). immiscible liquid of De Gennes (1985) paid particular attention to wetting and spreading, which culminated in a nearly the same density comprehensive account of capillary phenomena (de Gennes et al. 2010). Rowlinson & Widom Sessile drop: a liquid (2002) and others in the chemistry community provided a molecular perspective, and yet another drop on a solid substrate thread, with mathematical roots in the study of isoperimetric inequalities, has been provided by Finn (1999). Johns & Narayanan (2002) presented case studies in capillary stability, whereas Liquid bridge: a liquid enclosed by a Shikhmurzaev (2007) focused more on the flows that underlie capillary surfaces. In this article, we capillary surface focus on the mechanics of fluids subject to capillary instability. having two closed- Between 1994 and 2013, items published annually that list the terms sessile drop, liquid curve contact lines bridge, or liquid rivulet as a topic have surged nearly sixfold in number (Figure 1a). Nearly 3,000 located on separate publications have appeared in total. Using the terms capillarity, wetting, or spreading as alternative solid surfaces Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org HISTORICAL PERSPECTIVE Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. Popular public lectures by Boys in the 1890s featured soap-film experiments. Boys (1959 [1890]) attributed these experiments in capillarity to the published work of Savart, Plateau, Maxwell, Thomson, Rayleigh, Bell, and Rucker.¨ An entry on capillary action first appeared in the Encyclopedia Britannica in 1898 (Maxwell 1898), reflecting the mathematical physicists’ wide interest in the subject at that time (e.g., Schrodinger¨ 1915). About one-third of the 14-page entry speaks to the stability of capillary surfaces. The entry remains in the encyclopedia until 1926, at which time it redirects readers to an entry on surface tension, with an indication that Lord Rayleigh has taken over authorship, at least for the historical review part. Regarding the 1898 entry, Erle et al. (1970) pointed out that Maxwell confused volume and pressure disturbances in his discussion of the stability of the catenoid. 540 Bostwick · Steen FL47CH22-Steen ARI 1 December 2014 20:31 240 220 abc 200 180 Search topic V Liquid/fluid bridge 160 Sessile drop/droplet 140 Df Γ 120 ( ) 100 D(V) 80 Liquid σ 60 α Number of published items σ σ 40 ls sg 20 Ds Solid 0 p 1994 1998 2002 2006 2010 2013 Calendar year Figure 1 (a) Number of publications by year since 1994 found by searching on listed topics. (b) Definition sketch for a sessile drop of volume V α σ σ σ and equilibrium contact-angle , with liquid/gas ( ), liquid/solid ( ls), and solid/gas ( sg) interfacial energies per area. (c) Sessile drop response diagram for a pinned contact line: volume V against Laplace pressure p. topics shows an even stronger swell. Feeding this are streams of interest from new applications, mainly on the micro- and nanoscales. For droplets, these applications include nanoparticle assembly by droplet evaporation (Bigioni et al. 2006), the assembly of microparts via capillarity (Sariola et al. 2010), microassays for screening (Berthier et al. 2008), microencapsulation (Almeida et al. 2008), forensic bloodstains (Attinger et al. 2013), immersion lithography at high speeds (Harder et al. 2008), the cleaning of nanopatterned wafers (Xu et al. 2013, Shahraz et al. 2012), biomimetic surface design (Lafuma & Quer´ e´ 2003, Tuteja et al. 2007), and surface wettability engineering generally. Contact-angle measurement continues to be a principal means for the assessment of material wettability in many industries (Matsumoto & Nogi 2008, Seetharaman et al. 2013). For liquid bridges, applications include gravure printing (Dodds et al. 2012, Kumar 2015), colloidal aggregation (Kralchevsky & Denkov 2001), solid adhesion (Maugis 2000), bioinspired adhesion (De Souza et al. 2008, Slater et al. 2012, van Lengerich & Steen 2012), crystal growth in float zones (Lappa 2005), and extensional rheology (McKinley & Sridhar 2002), whereas, for rivulets, they include heat and mass transfer (El-Genk & Saber 2001). Inexpensive Contact angle: angle Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org access to high-speed imaging cameras may also be playing a role in the swell of interest. measured through a Simulations (Sui et al. 2014) and experiments (Milne et al. 2014) are crucial to the subject’s liquid, from the solid to the capillary surface; Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. development. Related, but also outside the scope of the present review, are thin films and coating a static or equilibrium flows (Oron et al. 1997); Marangoni flows (Matar & Craster 2009); the nonlinear dynamics of contact angle relates topological change that sometimes ensues from capillary instability (Eggers 1997, Paulsen et al. the liquid/gas, 2012); and software such as Surface Evolver (Brakke 1992), a versatile tool for predicting the liquid/solid, and reconfiguration of static surfaces (Collicott & Weislogel 2004). solid/gas surface Liquids that partially wet a solid support are of interest
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